Introduction Multiphase ows in porous media occur in various practical situations including oil recovery and unsaturated groundwater ow, see for insta

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1 Analysis of a Darcy ow model with a dynamic pressure saturation relation Josephus Hulshof Mathematical Department of the Leiden University Niels Bohrweg, 333 CA Leiden, The Netherlands hulshof@wi.leidenuniv.nl WWW: John R. King Department of Theoretical Mechanics, University of Nottingham Nottingham NG7 RD, United Kingdom john.king@nottingham.ac.uk Abstract We consider a new model for groundwater ow. The model diers from previous models in that the saturation-pressure relation is extended with a dynamic term, namely the time derivative of the saturation. The resulting model equation is of nonlinear degenerate pseudo-parabolic type. We give a rigorous analysis of travelling wave solutions and the local behaviour near fronts. Part of this analysis is then used in a formal asymptotic analysis of solutions with initial support in a half line. As an illustrative special case we also consider Heaviside initial data. AMS Subject Classication. 35K7, 35K65 Key Words and Phrases. Groundwater ow, Darcy's law, dynamic pressure relation, travelling wave solutions, fronts, asymptotic analysis, Heaviside initial data. Abbreviated Title. Darcy ow with dynamic pressure. We thank C.J. van Duijn for suggesting the mathematical model and encouraging our research in this direction. We gratefully acknowledge the support of the Dutch Organisation for Scientic Research (NWO) and the British Council. We also thank S.M. Hassanizadeh and A. Leijnse of the National Institute of Public Health and Enviromental Protection (RIVM) in Bilthoven, The Netherlands for stimulating discussions.

2 Introduction Multiphase ows in porous media occur in various practical situations including oil recovery and unsaturated groundwater ow, see for instance [6]. The mathematical modelling of such ows is therefore of great importance, the aim being to give qualitative and quantitative predictions of the saturation proles, based on knowledge about initial proles, in- and outow boundary conditions, and the governing equations modelling the ow. In the standard approach for two phase ows, such as oil-water or air-water mixtures, one combines the mass conservation equations and Darcy's law for the separate phases with a relation for the pressure dierence in the phases. Typically this relation is taken to be of the form p n? p w = p c (S w ); (.) where p n, p w are respectively the pressure of the non-wetting and the wetting phase, and where S w is the level of saturation of the wetting phase. The function p c (S w ) is called the capillary pressure. It is usually assumed to be a bounded decreasing function on the interval [; ] with some smoothness properties and satisfying p c () =. In the case of air (non-wetting) and water (wetting), the air pressure is often assumed to be atmospheric, so that p n is constant; henceforth we consider this case. Mass conservation of water is given + div q w = : (.) Here is the porosity of the porous medium and q w the volumetric water ux. Darcy's law reads q w =?K(S w ) grad p w ; (.3) where K(S w ) is the hydraulic conductivity, which is a nonnegative increasing function of S w [; ]. Note that we neglect gravity. Combining (.,.,.3) and restricting to one space dimension, we obtain a single nonlinear degenerate diusion equation for the saturation S w [9]: @x p c(s w (D(S w ); D(S w ) =?K(S w )p c(s w ): (.5) Equation (.4) is of great practical importance because it can actually be used to compute the saturation prole if the the constitutive functions K(S w ) and p c (S w ) are known. Usually these are determined from experiments and typically the coecient D(S w ) tends to zero as S w tends to zero. Replacing S w by u and taking as a model for the case of small saturations D(u) = u with >, we @u (u

3 which is regarded as the model equation for degenerate diusion. Equation (.6), which also arises in numerous other applications, has been subject of intensive research over the past decades, see for example [], [7], [], []. A striking property of (.6) is the occurrence of interfaces: solutions with compactly supported initial data remain compactly supported for all positive times. In terms of the groundwater model this corresponds to ows with a moving front separating the dry and the wet regions. Properties such as waitingtime phenomena, regularity and large time behaviour of the interfaces bounding the (expanding) support have been fully analysed. A drawback of this model is not so much the explicit choice of D(u) above, but rather the assumption that the capillary pressure is a function of the saturation only. This is generally considered by groundwater engineers to be untrue, measurements for wetting and draining exhibiting dierent pressure-saturation curves. To capture the dynamic aspects of two-phase ows more accurately, Hassanizadeh and Gray [] have recently proposed a new approach in which the pressure saturation relation (.) is replaced by an expression of the form p n? p w = p c (S w )? w : (L > ) The coecient L may be viewed as a capillary damping coecient. A similar relation was also suggested some time ago as an outcome of experiments by Stauer [6]. This changes equation (.4) @x p c(s w (K(S S w ): The corresponding model u ); where and are positive constants. Let us mention here that, as is the case with (.6), equation (.9) should be considered as a model for small saturations so that it is not unreasonable to take pure powers in the nonlinearities, which has the advantage of making the analysis more transparent and the calculations more explicit. Models closely related to (.9) also arise in other contexts; see, for example, [3]. A physically relevant initial boundary value problem consists of studying (.9) on an x-interval [; l], with nonnegative initial data u(x; ) = u (x) and zero ux lateral boundary conditions u u x + u u xt = on x = ; l. Assuming for the moment that 6= +, we may transform this problem into two equations which decouple the space and time derivatives, namely and u t +?(u w x ) x + w =? + u?+ = w for t ; u = u at t = ; (.)? + u?+ for x (; l); u w x = on x = ; l: (.) 3

4 It is clear from the theory of second order ordinary dierential equations that (.) is uniquely solvable in w if the function u is positive and smooth (in fact uniform bounds of the form < u M and Lebesque measurability suce if we consider weak solutions). Since the solution w depends smoothly on the choice of u, it follows that, with (.), the problem may be written as an ordinary dierential equation in an open subset of a Banach space of functions. In particular local well-posedness for piecewise continuous positive initial data is immediate. However, we are interested here in solutions with moving (or xed) fronts. In view of the degeneracy near u =, it is not yet clear what the proper denition of such solutions should be, and what kind of moving fronts one should expect. As in the case of (.6), one would expect such a solution to be obtained as the limit of smooth solutions u with initial data u (x; ) = u (x) +. Though positivity properties of the approximating smooth solutions may be obtained using maximum principle type arguments, there is, unfortunately, unless =, no comparison principle which allows us to conclude monotonicity in. Thus the existence of a (unique) limit as! is unknown. (For = well-posedness is immediate from (.) and (.). In this case also a strong maximum principle holds, so that nonnegative solutions instantaneously become strictly positive). For > and > we shall rst make (Section ) a detailed and rigorous study of possible travelling wave solutions and in particular of the behaviour of such travelling waves near u =. This amounts to a local analysis of the behaviour of general solutions close to a moving front; the global travelling wave proles are in fact always unbounded. This may seem unphysical at rst because we are thinking of saturations which should be bounded by denition; the point, however, is that one expects solutions near moving fronts to behave locally as a travelling wave. In other words, xing t and zooming in at the front by scaling u and x, the prole selects one of the (local) travelling wave proles. This is well known for (.6) where the speed of the selected prole coincides with the speed of the propagating front. In the case of (.6), for each >, there is, up to translation, only one travelling wave with a front for each (positive) speed. In the travelling wave analysis of (.9) we encounter what we will call generic and exceptional behaviours near u =, but there are also values of the parameters for which no travelling fronts exist. In order to give an interpretation of these travelling waves in terms of local front behaviour of general solutions, we shall carry out a formal asymptotic analysis (Section 3) of the limit solution obtained through the regularisation by setting u (x; ) = u (x) +. In this part of the analysis the limit solution is assumed to exist and behave well under perturbation. It will turn out that near a moving front the asymptotics select the exceptional behaviour of the solutions of the travelling wave equation near zero. Moreover, for values of the parameters for which there are no travelling wave solutions with a front, the asymptotics indicate that the front is in fact xed. Numerical experiments carried out by Leijnse [4] at the RIVM in Bilthoven, The Netherlands, also suggest that in some cases the front does not move at all and that the solution develops a rather steep standing prole. To illustrate this 4

5 behaviour, we also include an asymptotic analysis of solutions with Heaviside initial data (Section 4). In this analysis the global behaviour of the unbounded travelling waves will play a role. We mention that equation (.9) diers considerably from another third order extension of the porous medium equation involving mixed derivatives, see [3], where basically in the right-hand side of (.4) the quantity S w is replaced by S S w. The corresponding model equation would read, in @u (u + with, for example, (s) = s. The main mathematical dierence is that (.) allows a splitting similar to (.,.), but with w appearing only on the right hand side of (.). Therefore (.) is easier to analyse [8]. Equations such as (.9) and (.) are called pseudo-parabolic. They are also, in general, nonlinear and, for appropiate, and, degenerate. With, respectively, = = and (s) = s, both (.9) and (.) reduce to the same linear pseudo-parabolic equation. For relevant references the interested reader is referred to the book of Carroll and Showalter [7], and also to [4] and [5], where relaxation terms appear in a dierent context. Travelling wave solutions In this section we investigate for which values of and (.9) has advancing front solutions. As observed in the introduction this is a purely local analysis. We note that for a large class of equations including (.6), such an analysis has been an essential rst step towards a complete understanding of the full partial dierential equation and the appearance of interfaces, see [] for an exhaustive discussion. We set u(x; t) = U(); = x + ct (.) which gives for U() that cu = (U U + cu U ) ; (.) where primes denote dierentation with respect to. In this section the proles we consider are increasing in x and moving to the left. Thus we take c >. Since we are interested in moving fronts, we look for solutions with U() # and U() U () + U() U ()! as # >?; (.3) for some nite corresponding to the position of the front. The second relation in (.3) states that at the front the ux should be equal to zero (conservation of mass). In the rst part of the analysis we classify the solutions of (.) which satisfy (.3), in terms of both their local and their global behaviour. We then also identify the speeds for which there are solutions of (.) with U()! as #? and having an exceptional growth for ". This will be 5

6 needed for the asymptotic analysis of (.9) in Section 3. Therefore we choose not to scale c out of the equation, even though this is possible if 6=. Since the equation (.3) is autonomous we may, without loss of generality, take =. Using the boundary conditions (.3) we integrate (.) to give which we rewrite as the system, U + c U? U = U? ; (.4) U = V ; V = U?? c U? V = U? (U?? V ): (.5) c Let V + = f(u; V ) : U > ; V > ; V < cu? g and V? = f(u; V ) : U > ; V > ; V > cu? g. Note that, assuming U = V >, dv du = U? V We shall also use a number of related equations, namely Z = U? U ; s = log U : W = (U ) : Y =? (U? ) : A = Y =? U? : (.6) c dz ds + (? )Z = e(?)s Z ( c? Z); (.7) dy ds dw ds = W?? W? c e(? )s ; (.8) = (?? )Y? c + e(?)s Y ; (.9)? (U? ) : da ds = A( A c? + + )? e(?)s A 3 ; (.) In terms of Z we have that V + = fz > c g and V? = fz < c g.. Advancing fronts for < < By straightforward phase plane analysis we see that tracing back orbits in V + there are three possibilities: (i) U! u > and V! as #, (ii) U! and V! as #, or (iii) (U; V ) comes out of V?. Orbits in V? correspond to concave solutions U, so clearly, by shifting, these have U # as #. It is easily seen from integrating (.6) that for + this happens with unbounded V = U, while V = U remains bounded for < +. We call these solutions generic fronts because nearby solutions of (.3) also have fronts with the same local behaviour; we denote these -parameter families of solutions by U g. Since both (i) and (iii) can clearly occur and both are stable with respect to small changes, it follows that there is at least one exceptional solution to (.5) which satises (ii). In fact we shall see that, up to a shift in, only one such solution exists, and that it satises U # and U # as # ; we shall call such solutions exceptional fronts and denote them by U e. 6

7 .. The generic fronts for < < As already stated, we have that all solutions in V? come from the positive V - axis if < + and in fact they are parametrised by v = V (). Thus we have U g () v as # if < + : (.) If > + all the generic fronts are concave near U = with U unbounded. For a more detailed analysis of these solutions we employ the Z-equation (.7). Introducing P = e (?)s, we rewrite (.7) as P dz ds = (? )P Z + Z ( c? Z); P dp ds = (? )P ; (.) the P -factor in front of the s-derivatives may absorbed into a new independent variable. In the (Z; P )-plane the generic fronts come out of the origin. Using polar coordinates Z = r cos ', P = r sin ', we nd critical angles (restricting to P and Z ) ' = ; ' = ; ' = arctan c(?? ) : (.3) Linearising we nd that the rst two are saddle points but the corresponding unstable manifolds are just the positive Z-and P -axis. The third angle however is an unstable node. It follows that all generic fronts correspond to solutions with Y = U?+ U = P Z! c(?? ) : (.4) Returning to the Y -equation (.9) and writing we nd Y (s) = X(s) = e (??)s? K + Z s U g ()???? + X(s); (.5) c(?? ) c? e (?)a + X(a)da = (.6) c(??) Ke (??)s c(?? ) + e (?)s + O(e (?3?)s ) as s!?; (.7)? where K is an arbitrary constant. In particular we thus have? as # if > + : (.8) Finally we consider the case = +. For Y = V = U (which still goes to innity) we have This gives dy ds =? c + e(?)s Y (s)!? c as s #?: (.9) U g ()? log as # if = + : (.) c 7

8 .. The exceptional front for < < Here we consider orbits in the open rst quadrant of the (U; V )-plane which come out of the origin. These orbits can only lie in the set V +, so they have Z > c. We rst consider <. Writing the Z-equation (.7) as the system dz ds = (? )Z + QZ ( c? Z); dq ds = (? )Q; (.) these orbits come out of innity along the positive Z-axis. Thus in (.6) the rst term on the right-hand side dominates. It follows that W = U b? V goes to a positive constant, and the W -equation (.8) in fact gives that W = U b? V = (U )! r? as U # : (.) Writing (.8) as the system dw ds = W?? W? Q c ; dq ds = (? )Q; (.3) the corresponding orbit comes out of the saddle point (W; Q) = ( q? ; ) and is therefore unique. In particular we nd that the exceptional front is unique and satises U e () p (? ) as # if < : (.4) Next we consider >. In terms of Z, the exceptional front must have the property that Z > =c and, by inspection of (.7) and (.), that Z! =c as s!?, so that U? U! c. Introducing as new independent variable, we nd Writing we have R(t) =??? t Z t =?? e(?)s =? U? ; (.5) dz dt =?? Z? t + Z (? Z): (.6) c Z(s) = + R(s); (.7) c Z ( + t ) +?? e c d? ( +? t )?? e c F (R(t + ))d; (.8) ). Using a contraction argument, only one solution of where F (R) = R (R + c this integral equation with R(?) = is seen to exist, and it satises R(t)?? c? t as t!?; (.9) 8

9 so that, in terms of U, U e () c as # if > : (.3) Finally for the borderline case = we have that the Z-equation (.7) reduces to dz ds = (? )Z + Z (=c? Z); (.3) and the exceptional front corresponds to the unique (stable) positive zero of the right-hand side of (.3). Thus we nd the exact formula U e () = q c + 4c +?..3 Behaviour of the fronts for < < as! if = : (.3) For the sake of completeness we describe here what happens to the above fronts as becomes large. Inspection of the (U; V )-phase plane gives that all solutions enter the set V +, meaning that Z becomes larger than =c. For = it is clear from (.3) that Z tends to the positive zero of the right-hand side of (.3), whence all fronts have U() q c + 4c +? as! if = : (.33) If > the linear term in (.7) dominates the nonlinear term so that Z(s) K exp (? )s. In terms of V we have that V goes to a positive constant v. This can be seen directly from integrating the V -equation (.6). Thus we have U() v as! if > ; (.34) where v determines the solution completely. If =, the W -equation (.8) for W = V = U reduces to so that dv ds = V? c e(?)s ; (.35) U() p log as! if = : (.36) The W -equation (.8) implies immediately that all solutions have U() p as! if < < : (.37) (? ) Finally we see from the Z-equation (.7) that U() c as! if < : (.38) 9

10 . Advancing fronts for = For = the V -equation (.6) is separable and can be solved implicitly. We nd cu + c log ju? cj +? U? = C if 6= ; (.39) cu + c log ju? cj + log U = C if = ; (.4) where C is an arbitrary integration constant. In addition U() = c is always an explicit solution... The generic fronts for = As before we have where v (; ) determines the solution, U g () v as # if < ; (.4) U g ()??? c? as # if > ; (.4) and U g ()? log as # if = : (.43) c.. The exceptional front for = We have and U e () = c if ; (.44) U e () p as # if < : (.45) (? ) In both cases the exceptional fronts are unique. Note that for exceptional no longer means that the orbit comes out of (U; V ) = (; ). It is now exceptional in the sense that it is the only solution coming from the positive V -axis. For = = the exceptional front is given by U e () = c + c (e? c? ) =? 3 6c + 4 4c? : (.46)..3 Behaviour of the fronts for = as! We have for large -values that U ()! c if ; (.47) and U ()! v (; ) if > : (.48)

11 .3 Advancing fronts for > In this range, for which V + and V? are separated by a monotone decreasing curve asymptotic to the two axes, we encounter an unexpected phenomenon:.3. No advancing fronts for > and < We see from (.) that for < every orbit comes out of the set where Z > =c. In terms of (.5) this means that every orbit comes out of V +. In particular V is bounded. But then we have, since >, that dv=du U? =V as U!. For this implies that every orbit comes from the positive U- axis. Thus we conclude that for < every increasing solution of (.4) begins at a positive minimum, so that no advancing fronts exist. 4 3 V U Figure. The U; V -phase plane for (.5) with = 3, = 5 and c =..3. Critical wave speed for > and = For = the Z-equation reduces to (.3), which, since >, changes type when c crosses the value p. Above this value the analysis is the same as?

12 for <. In terms of U we have c > p? On the other hand we have c < p? ) no advancing fronts exist: (.49) ) U g () and U e () = q c? 4c +? as # q : (.5) c + 4c +? Both right-hand sides are exact solutions of (.4) and correspond, respectively, to the unstable and the stable positive equilibria of (.3). Finally we have c = p? ) U g () p? as # and U e () = p? ; (.5) so that in the leading order term there is no dierence between generic and exceptional solutions. This has consequences for the asymptotic analysis of the solution with Heaviside initial data, see Section Generic and exceptional fronts for > and < Since < implies <, we have again from (.) that every orbit comes out of V +, but now these orbits can also come from the positive V -axis (generic case) or from the origin (exceptional case). Thus we have U g () v as # if < : (.5) Using the W -equation (.8) the exceptional front has U e () p as # : (.53) (? ).3.4 Generic and exceptional fronts for > and > In terms of (.), the solution can now come out of the origin (Z; P ) = (; ) (generic fronts), out of the saddle-node (Z; P ) = (=c; ) (exceptional front), or from (Z; P ) = (; p) with p > (when U() has a positive minimum). The analysis of the generic case is similar to that with < and > + and leads to U g ()???? c? as # ; (.54) while the exceptional case is similar to that with < and > and leads to U e () c as # : (.55)

13 All fronts are contained in the set V? V U Figure. The U; V -phase plane for (.5) with =, = 6 and c =..3.5 Behaviour for > and large For > the behaviour of solutions of (.5) again depends on but also, in some cases, on the initial data. Let us rst observe that all solutions are (eventually) in V?. Thus they become concave with U() " as ". From the V -equation (.6) it is clear that, since >, > + ) 8v (; ) 9! solution with U ()! v as! ; (.56) while for + all solutions have the property that V () = U ()!. We rst continue with the case that > +. If we take = everything follows again from the reduced Z-equation (.7). We obtain = and c > p? ) U ()! v (; ) as! : (.57) For subcritical values of c we obtain as! two generic cases and one 3

14 exceptional case: = and c < p? ) U ()! v (; ) (generic); or U() = q (exceptional); c? 4c +? or U() q (generic): (.58) c + 4c +? For the critical wave speed we have as! = and c = U() = p? ) U ()! v (; ) (generic); p (exceptional); or U() p (generic); (.59)?? while for > we nd the same picture as for = with c supercritical: > ) U ()! v (; ) as! : (.6) This is easily seen from the Z-equation (.7) in which the linear term dominates. Next we consider + < <. Here we nd a similar picture as for = with c subcritical. We use (.), which we may also write as a system, da ds = A(A c? + + )? P A3 ; dp ds = (? )P: (.6) From (.6), combined with (.), we obtain that as s!, positive solutions of (.) either go to zero (generic), to c(?? ) (exceptional) or to innity (generic, corresponding to Z(s)! c ). Respectively, in terms of U, as! we have or U()? + < < ) U ()! v (; ) (generic);??? c (exceptional); or U() c (generic): (.6) Finally we consider +. It now follows that every solution has A(s)! as s!. In terms of Z, Z(s)! c. Thus we nd that the behaviour for large is independent of the initial conditions: + ) U() c as! : (.63) 4

15 .4 Advancing fronts with a positive limit In this section we consider solutions of the full travelling wave equation (.) with the property that U()! as!?: (.64) These solutions will play an important role in Section 3. Integration of (.) now yields or, written as a system, or U + c U? U = U?? U? ; (.65) U = V ; V = U? (U?? U V c dv du = U? V U ); (.66)? U? : (.67) c Clearly we have that for positive c the solution U = U of (.) with (.64) is given by the unique orbit coming out of the saddle point (U; V ) = (; ) into the rst quadrant of the (U; V )-plane. This orbit corresponds to the positive eigenvalue r + = + (c) =? c + + ; (.68) 4c and satises V + (c)(u? ) as U! : (.69) For ven c, U is therefore unique up to translations in. The leading order behaviour for large of increasing solutions of (.66) is exactly the same as for (.5), because in all cases U()! so that the extra term is of lower order. In all cases where there is only one large behaviour, this evidently also applies to U. For those values of and with more then just one kind of generic large behaviour, we can expect the existence of an exceptional speed c = c for which U () is exceptional..4. Exceptional speeds for > and + < < We recall that for this range there were no advancing fronts. The exceptional solution we construct plays a key role in the asymptotic analysis given below. The solution U = U ;c corresponds to a solution V (U) = V ;c (U) of (.67). It is clear that this function increases as c increases. To show that an exceptional speed exists, we rst prove that for large and small c the large behaviours are generic and are dierent. We then conclude that in between there is a value c = c for which the large behaviour is exceptional. We rst oberve that the Z-equation (.7) and the A-equation (.) now read dz ds + (? )Z = e(?)s Z ( c? (? e?s )Z); (.7) 5

16 and da ds = A(A c? + + )? e(?)s (? e?s )A 3 : (.7) For c! equation (.67) reduces to the limit equation so the limit solution is V dv du = U? U ; (.7) V ;(U) = (? )U?? (? )U +? (? )(? ) ; (.73) and in particular V ; () >. Using the smooth dependence on the parameter =c of the unstable manifold of (U; V ) = (; ) of the system (.66), it follows that V ;c (U)! V ; (U) uniformly on bounded intervals of the form f U Mg. Since the second term on the right-hand side of (.67) is integrable with respect to U, the convergence is also uniform on f U g. This yields for c suciently large a positive limit v ;c, so that V ;c (U)! v ;c > as U!, and thus U ;c()! v ;c > as! ; (.74) which is one the two generic behaviours. Next we show that by making c small we obtain the other generic behaviour for the orbit coming out of (U; V ) = (; ). In view of the fact that the orbit is bounded by V c(? )?? (which is the maximum V -value in fu > g in the (U; V )-plane along the curve where V = ), we employ the scaling V = c, = c, which changes the system (.66) into _U = c ; _ = U? (U?? U ): (.75) Thus we can consider the limit system with c = (in the fast variable only), which has a one-dimensional manifold of critical points given by M f = U? g; (.76) U which is easily seen to be normally hyperbolic in the sense of singular geometric perturbation theory, see for instance [5]. This implies that in a compact neighbourhood of (U; ) = (; ) there exists for every suciently small c > a onedimensional invariant manifold M c of (.75) which converges to M as c!. More precisely, M c is of the form M c = f = c (U)g and c (U)! (U? )U? as c!. Clearly, M c can only be the unstable manifold of (U; ) = (; ). Also there is no restriction on the right-hand side boundary of the U-interval on which c is dened, so we may take it as large as we want. It is not hard to see that, in terms Z(s), this means that we can keep Z(s) near =c for as large an s-value as we like, implying that from there on the right hand side of (.7) will force Z(s) to converge to =c. This corresponds to the other type of generic behaviour. 6

17 .4.3 V U Figure 3. The isocline V = and the orbit coming out of (; ) in the U; V -phase plane for (.66) with = 3, = 5 and c = (i.e. c large). 7

18 ..8.6 V U Figure 4. The isocline V = and the orbit coming out of (; ) in the U; V -phase plane for (.66) with = 3, = 5 and c = (i.e. c small). In view of the dierent generic behaviours for large and small c, it follows that there must exists a speed c = c for which U ;c has exceptional behaviour for large, meaning that U ;c ()???? c as! : (.77) The monotonicity with respect to c in (.67) implies that V ;c (U) is increasing in c and hence the corresponding A ;c (s) is decreasing in c. Since exceptional large behaviour corresponds to the unique orbit with A(s)! c(?? ), the limit being increasing with respect to c, the exceptional speed c for which A ;c (s)! c (?? ), is unique. 8

19 .4. Exceptional speeds for > and = Here the Z-equation reduces to dz ds =?(? )Z + Z ( c? (? e?s )Z)?(? )Z + Z (? Z); (.78) c where Z(s)! as s #. The exceptional speed coincides with the critical speed: c = p? : (.79) This is the only value of c for which U ;c selects the exceptional large behaviour. Indeed, the third order term prevents Z(s) from going back to innity, and the inequality (.78) keeps Z(s) above the zero's of?(? )Z + Z ( c? Z). Consequently it follows for c > c (when there is noqexceptional behaviour) that Z(s)!, while for c < c we have Z(s)! c + +?, which is one of the generic behaviours. For c = c it follows that Z(s)! p?, but only with an algebraic convergence rate..5 Advancing fronts, summary In the table below we list the local behaviours near zero of solutions of (.4), and the large behaviours of increasing solutions of (.4) and (.65). In all cases where there are three large behaviours, the second one listed is exceptional. For + < there is an exceptional wave speed c for which there is a unique solution of (.65) with U()! as!? having exceptional large behaviour. For = this exceptional speed coincides with the critical speed. 4c 9

20 < exceptional generic large < U e p (?) p c + U g v U c = Ue U 4c +? g v U < <+ Ue c U g v U = + U e c U g c log( ) U +< < U e? c Ug????? c c U = Ue? c Ug? > Ue c Ug???? c = exceptional generic large < U e p (?) U g v p c + 4c +? p (?) p (?) p (?) U p log U v U c = U e = c U g c log( ) U c > U e = c U? U v g?? > exceptional generic large < U e p (?) U g v c U c + none none U c +< < none none U???? c U v = none none U v c > p? = c = p? = c < p? > U e U v Ue = p? Ug p? U U e = p U c + 4c +? g?p c? c Ug??? c p? p? U U v U 4c +? U U v c?p 4c +? p c + 4c +?

21 3 Asymptotic methods The preceding analysis has been rigorous but was limited to a very special class of solutions. We now commence with the study of more general solutions of (.9) by means of formal asymptotic methods. Nevertheless, this analysis will rely quite heavily on the travelling wave solutions discussed above. 3. Moving fronts In order to determine the appropriate moving boundary conditions for (.9), we consider the physically motivated regularisation whereby the initial data takes the form u(x; ) = G(x) + ; G(x) = for x > ; < << : (3.) Note that we now typically will have a movement to the right, so that any travelling wave appearing here will be reversed with respect to the travelling waves in the previous analysis. We assume that the solution u of (.9) with these initial data is suciently smooth and examine its behaviour as!. Here we expect a two-layer structure: an outer region on the left where u u as!, with u > satisfying (.9), and an inner region situated at the boundary of this outer region. We will search for a scaling of u and x which allows the leading order behaviour of the interior layer to match with the outer behaviour. Locating the inner region at we will thus have x = s(t; ) s (t) as! ; (3.) u = for x s (t): (3.3) We introduce inner scalings, with > to be chosen later, and assume that u = v; x = s(t; ) + z; t = (3.4) v v as! ; (3.5) (v in this section diers in meaning from that of Section ), (.9) then @z ): (3.6) Since > the terms containing -derivatives are negligible as!, so v ): We must choose to give a limit equation for v which is solvable under the condition that v! as z! +; (3.8)

22 and which is such that the growth of the solution as z!? matches the scaling (3.4), which requires that Three cases must be discussed separately. 3.. > v = O((?z) ) as z!?: (3.9) Here we need =, so the rst two terms in (3.7) balance and dominate the third. Hence we (v ) = ; so that giving _s (v? ) = ; (3.) v? _s z as z!?; (3.) as required to match. In terms of u and x this leads to the porous medium equation balance u (x; t) _s (t)(s (t)? x) as x " s (t): (3.3) The regularisation thus selects the exceptional front, leading to a uniquely specied solution; this will be true of the other cases also. 3.. < Here we cannot take = because then we would obtain in the limit for v the equation (v v zz) z =, which with v! as z! + gives only v. Thus we have to balance the terms in another way. The only possible choice turns out to be = ; (3.4) and the limit equation becomes Assuming _s 6= we and for < we =?? v??? v ): = v?? v? ; (3.6) + (? )(? ) ; (3.7)

23 (in the special case = there are obvious changes) so that v (? ) z as z!?; (3.8) which, in terms of u and x leads to the moving boundary condition u (x; t) This does not involve _s (t) or even the sign of _s (t). For =, however, we nd (? ) (s (t)? x) as x " s =?p r log(v ) + v? ; (3.) incompatible with the scaling, which requires linear growth in this case. For > we nd that v has linear growth, again incompatible with the scaling and therefore unmatchable. We thus deduce that there are no solutions with travelling fronts for <, which is consistent with our travelling wave analysis in Section = Now, with =, all three terms in (3.7) balance as! and we obtain upon integration _s (v? ) = _s v ; = : This is the traveling wave equation (.65) for fronts with a positive limit (with the change of notation U $ v ; c $ _s ; $?z). The analysis depends on and was performed in Section.4. For < < we only have to transcribe (.33) and obtain or, in terms of u, v (z)?z q _s + +? 4 _s as z!?; (3.) u (x; t) q _s + +? (s (t)? x) as x " s (t): (3.3) 4 _s Thus the exponents are the same as in the case of porous medium equation balance (3.3), but the coecient is dierent. The case > corresponds to Section.4.. We nd the same balance as in (3.3), but with the restriction that the speed of the front is subcritical: _s p? : (3.4) 3

24 In case of equality the generic solutions in (.5) are also allowed in the matching procedure, see also Section 4.4. Finally = leads again to the porous medium equation balance u (x; t) _s (t)(s (t)? x) as x " s (t): (3.5) 3. Fixed fronts for < Here we consider the remaining range of parameters <, for which the methods in Section 3. failed to give any moving fronts. It will turn out that the front is in fact a xed boundary, at which the no-ux u u = at x = has to be imposed on the outer solution u (x; t), which satises (.9) for x < with initial data u (x; ) = G(x). We assume here that this solution exists and that U(t) = lim u (x; t) > ; (3.7) x!? (in fact this relies on the local behaviour of G(x) as x!?, but it is certainly true if G() > ). We also dene W (t) = _U(t) +? + U(t)?+ ; (3.8) Now let u be the solution of (.9) with initial data (3.). We start by scaling the space variable and set x = X, where = ()! as!. Assuming that u U (X; t) as!, we obtain for U that = Integrating once we nd + @X + U = ; (the constant of integration must be zero as a consequence of the rescaling) and, integrating and matching it +? + U?+ = W (t); (3.3) again with the obvious change if = +. We now let X = s(t; ), with s(t; ) s (t) as!, denote the position of the edge of the support of U so that (3.3) is subject to the initial condition U = at X = s (t). To the left of this front we assume the leading order behaviour of u to be given by (3.3) whereas to the right we assume that u = o() as!. 4

25 3.. < < + When < +, the second term on the left-hand side of (3.3) is small near X = s (t), so (t); as X! s (t)? ; (3.3) which calls for a linear scaling of u and X near X = s(t; ), i.e. u = v; X = s(t; ) + z; (3.33) with v v as!. The third term in (.9) then dominates the second but balances the rst if we choose =? @z (v Imposing v! as z!? then yields and =?? (? )(? )?? v? + v? We have from (3.3) that v ): v? ; (3.35) z as z!?: (3.36) (? )(? ) U W (t) _s (t) (s (t)? X) as X! s (t)? ; (3.37) so matching with (3.36) implies that r (? )(? ) s (t) = Z t In the original (x; t) coordinates the `front' is given by and in the limit! we thus obtain a xed front. 3.. = W ()d: (3.38) x? s (t); (3.39) The main dierence from the previous section is that the scalings read x = X = s(t; ) + z; = (log(=))? : (3.4) 5

26 The formulae for v and s change accordingly. =?p? log v + v? ; v? p z(log(?z)) as z!?; (3.4) so that, again matching with (3.3), the above choice of follows. The `front' is given by < < Z x (log(=))? s (t) = (log(=))? t p W ()d: (3.4) When > +, the second term on the left-hand side of (3.3) is large near X = s (t), so that which calls for the + U?+ as X! s (t)? ; (3.43) u = v; X = s(t; ) +? z: (3.44) To obtain a balance in (.9) we must now choose to give the full travelling =? _s =? @z v ): Thus v + = ; (3.47) where we require a solution which has a behaviour for z!? consistent with the scaling (3.44). This is (apart from a change of notation v $ U, c $ _s, $?z) the equation (.65), for which the analysis in Section.4. gave the existence of a unique exceptional speed _s (t) = c for which (in terms of v ) we have v??? c z: (3.48)?? In terms of U it follows from (3.43) that U? c??? (s (t)? X) as X! s (t)? ; (3.49) and matching is thus accomplished if s (t) = c t: (3.5) In the original (x; t) coordinates the `front' is thus given by x? c t: (3.5) 6

27 3..4 = + Here equation (3.3) changes into so that (3.43) is + log U = W (t); log U as X! s (t)? ; giving U _s (s? X) log(=(s? X)): We employ almost the same scaling as in the case < < +, (3.53) u = v; x = s(t; ) + z; (3.54) which again leads to and (3.35) and (3.36). Matching with (3.5) now gives that so that the `front' is given by r =? (? )(? ) log(=); _s (t) = ; (3.55) x? log(=)s (t) =? log(=)r (? )(? ) t: (3.56) 4 Heaviside initial data We now consider initial data of the form u(x; ) = H(?x); (4.) where H is the Heaviside function and we assume that the solution is that obtained as a formal limit of solutions u with initial data u (x; ) = H(?x)+. The analysis in Section 3. has the surprising corollary that in the range < this limit solution does not move at all, in other words, u(x; t) = H(?x). For the remaining cases we shall look for a self-similar structure of the limit solution with Heaviside initial data as t! + which has a front behaviour consistent with the analysis in Section 3.. At this point we note that equation (.9) allows self-similar solutions of the form and u(x; t) = t? f(); = xt? (?) for 6= ; (4.) u(x; t) = exp(t)f(); = x exp(?t=) for = : (4.3) Such self-similar solutions of the full equation will play a role in the asymptotic analysis of small time behaviour in the range > max( + ; ). To determine the local self-similarity of the solution with Heaviside initial data, we rst examine the behaviour of u( + ; t) for < t <<. We use the decoupling of (.9), 7

28 see (.,.) with + log u if = +. Assuming the continuity of w we [u] +?? + [u?+ ] +? = if 6= + ; (4.4) +? + [log u]+? = if = + : [u] We hence obtain that [u] +? =? exp(?t) if =. At this stage we claim that for general and, the rate of change of u( + ; t) is much larger than that of u(? ; t) because >. Consequently, u( + ; t) t as t! for < + ; (4.6)? + u( + ; t) t log(=t) as t! for = + ; (4.7)? u( + ; t) (?? t)? as t! for > + : (4.8) In the latter two cases our claim above follows directly from (4.4) and (4.5). In the rst it can readily be veried a posteriori; see Section 4. below. We now discuss these cases separately. 4. < min(; + ) We look for a self-similar form consistent with (4.6). The only possibility is which satises the balance In terms of f this leads to f() = u tf(); = xt?= ; s t = u ): (? ) ( ((? ))? ) ; = (? + ) = : (4.) For > however this is not consistent with the local behaviour found in Section 3. and an additional region is then needed with with scalings x = s(t) + t (?) (?) ; u = t?? ; ( < < + ); (4.) with leading order balance again being of travelling wave : (4.3) It then follows from Section..3 that the exceptional travelling wave solution matches the outer and the inner behaviours. Notice that the accumulative amount of mass that ows through x = from left to right is of order t +=, so that in particular u(? ; t) = + o(t) as t!. This justies our claim that u(? ; t) changes much more slowly than u( + ; t). 8

29 4. = +, < For < the dierences from the analysis in Section 4. are that with and u t log(=t)f(); = x(t log(=t))?= ; s (t log(=t)) = ; (4.4) x = s(t) + t (log(=t))? = ((? )) = =; (?) ; now give the scalings in the travelling wave region. 4.3 > max(; + ) u = t(log(=t))?? (4.5) The local behaviour (4.8) leads to (4.), with boundary conditions? f() = (?? )? ; f() (? ) (? ) as!? ; (4.6) where is one of the unknowns. We conjecture that there is a unique solution f(). 4.4 =, > As in Section 4.3 we are led to (4.), i.e. to u(x; t) t f(); = x=t; (4.7) which has to satisfy q f() = (? ) and f() (?? 4(? ) (? ) )(? ) (4.8) as!?. Condition (3.4) implies that p?. Since the right hand side above is in fact a solution of the corresponding full similarity equation, but satises the wrong boundary condition at =, it follows that = =( p? ): (4.9) Consequently the local front behaviour is non-generic in this case, but see however the nal remarks of Section.3.. Here too we conjecture the existence of a unique solution f(). 4.5 =, = Here we nd u(x; t) t log(=t)f(); = x=(t p log(=t)); (4.) leading to the third order term being dominating in (.9), giving f() =? and =. 9

30 5 Discussion Our analysis has shown that basically there are three possible fronts for (.9). One is of porous medium type ( > ), and another (new) one has local behaviour in which the second order term in (.9) is irrelevant ( < min(; )). One of the key results of the foregoing analysis is that there is a third range of (; ) for which (.9) exhibits xed fronts and u develops a discontinuity. This represents a new phenomenon in the current context and is consistent with certain experimental observations which formed part of the motivation for this work [4]. Here we conclude with a short discussion of two issues related to this aspect. 5. Other models An alternative model closely related to (.) is (cf. [3], We briey discuss the more + u+ ): (5.) @u ); with >, >, which has as special cases both (.9) ( = ) and (5.) ( = ). For > solutions to (5.) have travelling wave fronts of porous medium type and there is little qualitative dierence between solutions for dierent. For <, we investigate the possibility of xed fronts by rst seeking a local balance of the form u A(s(t)? x) as x! s(t)? ; (5.3) from which we nd that A = =(( +?)). For a moving front we require that < +, while for + < < ; (5.4) we conclude that xed fronts occur. For =, (5.4) implies < < (we omit discussion of borderline cases here) whereas (5.4) cannot be satised for =. We thus in particular conclude that (5.) cannot exhibit fronts which are xed for all time and this is a very important qualitative dierence between (.9) and (5.). 5. The porous medium equation limit Here + u ); where << will be chosen shortly and = O(), subject to (3.) and with < <. If we take the limit! followed by! we obtain, as the 3

31 leading order outer problem for u, the porous medium equation (.6) with the usual condition (3.3) at the moving front. However, it follows from the analysis in Section 3. that if we take the limits in the opposite order (! followed by! ) we obtain (.6) subject not to (3.3) = at x = ; (5.6) taking x = to be the initial location of the right-hand front. Thus (.6) with (3.3) is structurally unstable to perturbation by the term and here we briey discuss (omitting almost all of the details) the transition between (3.3) and (5.6). 5.. < < + Here the appropriate choice to describe the transition is =?. There are inner regions with scalings x? s(t; ) = O(? ), u = O() and x? s(t; ) = O(? ), u = O(). Matching to these (cf. Section 3..) shows that (.6) is to be solved subject to u?+ = p (? + ) p( _s ; u?? =? _s at x = s : (5.7) Hence in the limit! we recover (3.3) while for! we obtain _s! and thus (5.6) < < Here we must take =? and there are two cases to consider. With two inner layers with scalings x? s(t; ) = O(? ), u = O() and x? s(t; ) = O( ), u = O(), we obtain (in a somewhat similar fashion to Section 3..3) the matching =? _s =?c = p at x = c t= p ; (5.8) while with a single inner layer of scaling x? s(t; ) = O( ), u = O(), we arrive at (3.3). It follows from Section.4. that successful matching into the travelling wave layer in the latter case requires that _s < c = p, giving the required generic behaviour. We conclude that (.6) is now to be solved subject =? _s ; _s = c = p ; u > at x = s if t < t c ; u! ; _s ; _s < c = p as x! s? if t > t c : (5.) Thus at some nite time t c, u (s (t); t) drops to zero and the standard porous medium balance (3.3) is then recovered (in contrast to (5.7) where it is obtained 3

32 only as t! ). For t < t c, the inner region travelling front is exceptional, and hence determines _s (but u (s (t); t) is then free) whereas for t > t c it is generic, _s being determined instead by the outer problem with u (s (t); t) =. The behaviour at the front may, however, switch back; this will depend on the initial data. We note that (5.9)-(5.) represent novel moving boundary conditions for the porous medium equation. References [] S.B. Angenent, Analyticity of the Interfaces of the Porous Media Equation after the Waiting Time, Proc. A.M.S., : , 988. [] D.G. Aronson, The Porous Medium Equation, in: Some Problems in Nonlinear Diusion, eds. A. Fasano & M. Primicerio, Lecture Notes in Math., CIME Foundation Series, Springer Verlag No.4, 986. [3] G.I. Barenblatt, V.M. Entov & V.M. Ryzhik, Theory of Fluid Flows Through Natural Rocks, Kluwer, Dordrecht-Boston-Leiden, 99. [4] G.I. Barenblatt, M. Bertsch, R. DalPasso, V.M. Prostokishin & M. Ughi, A mathematical model of turbulent heat and mass transfer in stably stratied shear ow, J. Fluid Mech. 53: , 993. [5] G.I. Barenblatt, M. Bertsch, R. DalPasso & M. Ughi, A degenerate pseudoparabolic regularization of a nonlinear forward-backward heat equation arising in the theory of heat and mass exchange in stably stratied turbulent shear ow, SIAM J. Math. Anal. 4: , 993. [6] J. Bear, Dynamics of Fluids in Porous Media. Elsevier, New York 97. [7] R.W. Carrol & R.E. Showalter, Singular and Degenerate Cauchy Problems, Academic Press, 976. [8] E. Dibenedetto & M. Pierre, On the maximum principle for pseudoparabolic equations, Indiana Univ. Math. J., 3: 8-854, 98. [9] C.J. van Duijn, J. Molenaar & M.J. de Neef, The Eect of Capillary Forces on Immiscible Two-Phase Flow in Heterogeneous Porous Media, Transport in Porous Media, : 7-93, 995. [] B.H. Gilding & R. Kersner, The Characterization of Reaction-Convection- Diusion Processes by Travelling Waves, J. Di. Equ., 4(): 7-79, 996. [] S.M. Hassanizadeh & W.G. Gray, Thermodynamic Basis of Capillary Pressure in Porous Media, Water Resources Research, 9: , 993. [] J. Hulshof, Similarity Solutions of the Porous Medium Equation with Sign Changes, J. Math. Anal. & Appl. 57: 75-, 99. 3

33 [3] J.R. King, The isolation oxidation of silicon, SIAM J. Appl. Math. 49: 64-8, 989. [4] A. Leijnse, Private communication, 995. [5] J. Smoller, Shock Waves and Reaction-Diusion Equations (nd edition), Springer-Verlag, Berlin Heidelberg New York, 994. [6] F. Stauer, Time dependence of the relations between capillary pressure, water content and conductivity during drainage of porous media, International Association of Hydraulic Research (IAHR) Symposium on "Scale Eects in Porous Media", Thessaloniki, Greece, Aug 9-Sept, 978. [7] J.L. Vazquez, An Introduction to the Mathematical Theory of the Porous Medium Equation, in: Shape Optimization and Free Boundaries, ed. M. C. Delfour, Mathematical and Physical Sciences, Series C, 38: , Kluwer, Dordrecht-Boston-Leiden,